International Journal of Network Security, Vol.20, No.6, PP.1206-1214, Nov. 2018 (DOI: 10.6633/IJNS.201811 20(6).21) 1206 An Improved AES S-box Based on Fibonacci Numbers and Prime Factor Kamsiah Mohamed1, Fakariah Hani Hj Mohd Ali1, Suriyani Ariffin1, Nur Hafiza Zakaria2 and Mohd Nazran Mohammed Pauzi3 (Corresponding author: Kamsiah Mohamed) Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA1 40450 Shah Alam,Selangor, Malaysia Faculty of Science and Technology, University Sains Islam Malaysia2 71800 Nilai, Negeri Sembilan, Malaysia Faculty of Engineering and Life Sciences, Universiti Selangor3 45600, Bestari Jaya, Selangor, Malaysia (Email: [email protected]) (Received Aug. 28, 2017; revised and accepted Dec. 18, 2017) Abstract group of bits known as blocks to transform a fixed length block of plaintext message blocks into cipher text blocks This paper emphasises the study on ways of constructing of the same length. Since 1970, block cipher design and the substitution boxes (S-boxes). To improve the strength analysis have been widely studied culminating in the se- of block cipher, a new proposed substitution box for sym- lection of Rijndael [8] as the new Advanced Encryption metric key cryptography was designed based on Fibonacci Standard (AES) in 2001 [5]. Thus, a modern block cipher numbers and prime factor. This new security approach was designed based on the AES substitution box (called was designed for better security of block ciphers. The S-box) to substitute blocks of input bits to a set of out- level of security S-box was evaluated based on the cryp- put bits. S-box is a critical part of any block cipher that tographic properties such as balance criteria, nonlinear- provides the primary source non-linear [12, 16]. ity, correlation immunity, algebraic degree, transparency This paper proposes a design of secure symmetric en- order, propagation, number of fixed points and opposite cryption S-box to improve the existing S-box. The de- fixed points, algebraic immunity, robustness to differen- sign and characteristics of S-box in a block cipher are tial cryptanalysis, signal to noise ratio (SNR) Differential the central measures of resistance against all adequately Power Analysis (DPA) as well as confusion coefficient. high nonlinearities [9]. The confusion and diffusion prop- The AES S-box and the new proposed S-box were anal- erties are needed to build a strong encryption algorithm ysed to verify the cryptographical security of the S-box. as suggested by [37]. However, there are some problems Result showed that the new proposed S-box using the Fi- addressed in the process of the designing of a new S-box. bonacci numbers and prime factor possessed good cryp- The two sets of problems arise from the selection of an S- tographic properties compared to the AES S-box. box before its cryptographic use can be considered secure. Keywords: Block Cipher; Cryptography; Fibonacci; S-box The first problem is related to the design (or search) of a good S-box while the second problem is in the verifica- 1 Introduction tion of a given S-box as one cryptanalytic technique [2]. Hence, constructing secure S-boxes to use them in dif- Cryptography is an important part of information secu- ferent cryptosystems for increasing their security is the rity that covers the investigation of algorithms and pro- current study problem [17]. S-box design is usually the tocols for secure information. Within the advancement most important task while designing a new cipher [7]. of technology, the design of cryptographic algorithm is The design of the new S-box is an important concern often enhanced to ensure that information is secure. In in creating new and more secure cryptosystems [11]. The terms of security, it is always a question of whether or not disadvantages of S-box design are the limitations that these algorithms are secure enough to protect informa- make it vulnerable and insecure [1, 18]. Currently, there tion. Block ciphers are the most prominent and impor- are no algebraic procedures that can give the preferred tant elements to provide high level security. Generally, and complete set of properties for an S-box [33]. Thus, block cipher is a deterministic algorithm on fixed length there has been a lot of attention on redesigning, recre- International Journal of Network Security, Vol.20, No.6, PP.1206-1214, Nov. 2018 (DOI: 10.6633/IJNS.201811 20(6).21) 1207 ating or renewing the design and implementation of the crypting message. original AES S-box. In another study [34], it was shown that the content Based on previous studies, there are various techniques of the original message were changed to the ciphertext by used to construct the standard AES S-box such as linear- taking each character from the message and converting it transform and non-linear function [39], fractional linear based on the Fibonacci numbers. Based on these previous transformation [19], branch numbers [36], affine trans- studies, understanding the role of the Fibonacci numbers formation [6, 41] and the network RFWKIDEA32-1 [26]. may be a key to increase the performance of block cipher In another study, it was shown that Fibonacci number in cryptosystems. can make secure communication from cryptanalysis at- tacks [35]. This technique can fulfil the requirements for communication such as capacity, security and robustness 3 AES S-box in Cryptography to secure data transmission over an open channel. Recent Substitution is a nonlinear transformation that makes the studies proved that the performance of encryption and confusion of bits. It is often considered as a look-up ta- decryption algorithm using Fibonacci number is faster ble, which uses several byte substitution transformations than symmetric algorithms [38] and RSA algorithms [14]. in the key expansion routine to perform a one-for-one sub- These studies demonstrated that the performance of en- stitution of a byte value. An n × m S-box is a mapping cryption and decryption algorithm can be increased using from n input bits to m output bits, S : f0; 1gn ! f0; 1gm. Fibonacci numbers. However, no study has addressed Fi- Basically, an S-box is a set of m single output Boolean bonacci technique and prime factor to construct the AES functions combined in a fixed order. There are 2n inputs S-box. In this paper, Fibonacci numbers and prime factor and 2m possible outputs for an n × m S-box. Generally, were used to improve the original AES S-box. an n×m S-box, S, is represented as a matrix of size 2n×m This paper is organized as follows: In Section 2, previ- for each m-bit entry. An n × m S-box is a bijective S-box ous studies on Fibonacci numbers are reviewed. Section 3 where each input is mapped to a dissimilar output entry briefly describes the AES S-box in cryptography, the Fi- and all possible outputs are presented in the S-box. bonacci numbers and prime factor. Comparison between In 2001, the National Institute of Standards and Tech- the properties of Boolean functions of our new proposed nology (NIST) announced the AES as a new standard S-box and the AES S-box is explained in Section 4. Fi- to replace the Data Encryption Standard (DES). This nally, conclusion is presented in Section 5. standard indicates that the Rijndael algorithm is gener- ally utilized as a part of numerous cryptographic appli- cations. It was designed to handle additional block sizes 2 Review on Fibonacci Numbers and key lengths 128, 192 and 256 bits. The 128 bits AES encrypted a 16 byte block using a 16 byte key of 10 en- In the field of cryptography, numbers play an important cryption rounds. The value of each byte in the array is role in different theoretical and practical applications. substituted according to a look-up table. Cryptosystems rely on the assumption that a number of The Rijndael AES S-box is designed based on three mathematical problems are computationally intractable transformations. The S-box is generated by determining since they cannot be solved in polynomial time [29]. The multiplicative inverse for a given number in Galois Field Fibonacci numbers are natural numbering system appro- GF(28) using the irreducible polynomial: priate for the development of each living thing. Many 8 4 3 1 studies have investigated on how Fibonacci sequence can m(x) = x + x + x + x : be observed in the real world. These numbers occur ev- The multiplicative inverse is then transformed as in Equa- erywhere in nature, ranging from the leaf arrangement tion (1): in plants, the structure of DNA as well as various pro- 0 portions in human face and structure of sea shells. One xi = xi ⊕ x(i+4)mod8 ⊕ x(i+5)mod8 ⊕ x(i+7)mod8 ⊕ Ci: (1) study has been conducted observing that the phyllotaxis Where xi is the bit i of the byte and the column vector Ci of plants follows the Fibonacci sequence [28]. is added with the value f63g or f01100011g. The affine A study by [23] showed that the structure of DNA and transformation element of the Rijndael AES S-box can be its organization pattern is a fractal. Then, [31] discovered expressed as shown in Figure 1. that the DNA gene-coding region sequences are strongly This affine transformation is the sum of multiple rota- related to the Golden Ratio and Fibonacci/Lucas integer tions of the byte as a vector. Figure 2 shows the original numbers. In another study, [3] examined the Fibonacci AES S-box represented here with hexadecimal notation. numbers can be seen in the structure of coronary arterial tree and that diseased atherosclerotic lesions in coronary 3.1 Construct AES S-box Using Fi- arteries follow the Fibonacci distribution.